Central sets theorem in arbitrary semigroup (Q6172293)

In the article, a generalised version in terms of matrices of the Central Sets Theorem is provided. The Central Sets Theorem, which can be seen as a joint extension of the Finite Sum Theorem and van der Waerden's Theorem, was originally introduced by Furstenberg. For a set \(A\), let \({}^AA\) be the set of all functions from \(A\) to \(A\), let \(\operatorname{fin}(A)\) be the set of all finite subsets of~\(A\), and for positive numbers \(m\), let \(A_m\) and \(A^m\) be the set of all \(m\)-tuples of \(A\) considered as a \(1\times m\)-matrix or as a \(m\times 1\)-matrix, respectively. The main theorem of the article states as follows: Let \((S,+)\) be a commutative semigroup and let \(B\) be a central subset of~\(S\). Then for any positive natural number \(m\) there are functions \(\alpha:\operatorname{fin}(({}^SS)_m)\to S\) and \(H:\operatorname{fin}(({}^SS)_m)\to \operatorname{fin}(S^m)\) such that that for any \(G,G'\in \operatorname{fin}(({}^SS)_m)\) with \(G\subsetneq G'\) we have \(\max H(G)<\min H(G')\), and whenever \(r\) is a positive integer, \(G_1\subsetneq G_2\subsetneq \ldots \subsetneq G_r \in \operatorname{fin}(({}^SS)_m)\) and \(A_i\in G_i\) (for \(1\leq i\leq r\)), we have: \[\sum_{i=1}^r\left(\alpha(G_i)+ \sum_{t\in H(G_i)} A_i\cdot t \right)\in B\]